Group #28290640 — subgroups

Subgroups — C₂³×C₄²×D₅₃ #28290640

Order 13568.

24030 conjugacy classes of subgroups; only the 13515 normal subgroups are shown.

normal subgroups only

This group has 24030 conjugacy classes of subgroups — too many to draw. Only its 13515 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₅₃	C₂	1	3	yes	maximal
2	C₂³×C₄×Dic₅₃	C₂	1	3	yes	maximal, complemented
3	C₂²×C₄²×D₅₃	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×D₅₃	C₂²	1	1	yes	characteristic
6	C₂⁴×Dic₅₃	C₂²	1	3	yes	—
7	C₂³×C₄×D₅₃	C₄	1	96	yes	direct product factor
8	C₂³×C₄×D₅₃	C₂²	1	84	yes	—
9	C₂²×C₄×Dic₅₃	C₂²	1	84	yes	complemented
10	C₂×C₄²×D₅₃	C₂²	1	448	yes	direct product factor
11	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	28	yes	complemented
13	C₂⁴×D₅₃	C₂×C₄	1	48	yes	—
14	C₂⁴×D₅₃	C₂³	1	14	yes	—
15	C₂³×Dic₅₃	C₂×C₄	1	48	yes	complemented
16	C₂³×Dic₅₃	C₂³	1	42	yes	—
17	C₂²×C₄×D₅₃	C₂×C₄	1	1344	yes	direct product factor
18	C₂²×C₄×D₅₃	C₂³	1	336	yes	—
19	C₂×C₄×Dic₅₃	C₂³	1	336	yes	complemented
20	C₄²×D₅₃	C₂³	1	512	yes	direct product factor
21	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	C₂×C₄	1	48	yes	complemented
23	C₂³×C₄×C₅₃	C₂³	1	42	yes	—
24	C₂×C₄²×C₅₃	C₂³	1	112	yes	complemented
25	C₂⁴×C₅₃	C₂²×C₄	1	24	yes	—
26	C₂⁴×C₅₃	C₂⁴	1	7	yes	—
27	C₂³×D₅₃	C₄²	1	256	yes	direct product factor
28	C₂³×D₅₃	C₂²×C₄	1	336	yes	—
29	C₂³×D₅₃	C₂⁴	1	28	yes	—
30	C₂²×C₄×C₅₃	C₂²×C₄	1	336	yes	complemented
31	C₂²×C₄×C₅₃	C₂⁴	1	84	yes	—
32	C₂²×Dic₅₃	C₂²×C₄	1	336	yes	complemented
33	C₂²×Dic₅₃	C₂⁴	1	84	yes	—
34	C₂×C₄×D₅₃	C₂²×C₄	1	2688	yes	direct product factor
35	C₂×C₄×D₅₃	C₂⁴	1	192	yes	—
36	C₄²×C₅₃	C₂⁴	1	64	yes	complemented
37	C₄×Dic₅₃	C₂⁴	1	192	yes	complemented
38	C₂³×C₅₃	C₂×C₄²	1	64	yes	complemented
39	C₂³×C₅₃	C₂³×C₄	1	84	yes	—
40	C₂³×C₅₃	C₂⁵	1	7	yes	—
41	C₂²×D₅₃	C₂×C₄²	1	896	yes	direct product factor
42	C₂²×D₅₃	C₂³×C₄	1	336	yes	—
43	C₂²×D₅₃	C₂⁵	1	8	yes	—
44	C₂×C₄×C₅₃	C₂³×C₄	1	336	yes	complemented
45	C₂×C₄×C₅₃	C₂⁵	1	24	yes	—
46	C₂×Dic₅₃	C₂³×C₄	1	336	yes	complemented
47	C₂×Dic₅₃	C₂⁵	1	24	yes	—
48	C₄×D₅₃	C₂³×C₄	1	768	yes	direct product factor
49	C₂⁴×C₄²	—	53	1	no	maximal, Sylow
50	C₂²×C₅₃	C₂²×C₄²	1	112	yes	complemented
51	C₂²×C₅₃	C₂⁴×C₄	1	42	yes	—
52	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
53	C₂×D₅₃	C₂²×C₄²	1	448	yes	direct product factor
54	C₂×D₅₃	C₂⁴×C₄	1	48	yes	—
55	C₄×C₅₃	C₂⁴×C₄	1	48	yes	complemented
56	Dic₅₃	C₂⁴×C₄	1	48	yes	complemented
57	C₂⁵×C₄	—	53	3	no	—
58	C₂³×C₄²	—	53	56	no	—
59	C₂³×C₄²	—	53	3	no	—
60	C₂³×C₄²	D₅₃	1	1	yes	characteristic, direct product factor
61	C₂×C₅₃	C₂³×C₄²	1	28	yes	complemented
62	C₂×C₅₃	C₂⁵×C₄	1	3	yes	—
63	D₅₃	C₂³×C₄²	1	32	yes	direct product factor
64	C₂⁶	—	53	1	no	—
65	C₂⁴×C₄	—	53	3	no	—
66	C₂⁴×C₄	—	53	96	no	—
67	C₂⁴×C₄	—	53	84	no	—
68	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
69	C₂²×C₄²	—	53	84	no	—
70	C₂²×C₄²	—	53	448	no	—
71	C₂²×C₄²	C₂×D₅₃	1	28	yes	direct product factor
72	C₅₃	C₂⁴×C₄²	1	1	yes	characteristic, complemented, Sylow
73	C₂⁵	—	53	48	no	—
74	C₂⁵	—	53	14	no	—
75	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
76	C₂³×C₄	—	53	48	no	—
77	C₂³×C₄	—	53	42	no	—
78	C₂³×C₄	—	53	1344	no	—
79	C₂³×C₄	—	53	336	no	—
80	C₂³×C₄	C₄×D₅₃	1	48	yes	direct product factor
81	C₂³×C₄	C₂²×D₅₃	1	42	yes	—
82	C₂×C₄²	—	53	336	no	—
83	C₂×C₄²	—	53	512	no	—
84	C₂×C₄²	C₂²×D₅₃	1	112	yes	direct product factor
85	C₂⁴	—	53	256	no	—
86	C₂⁴	—	53	336	no	—
87	C₂⁴	—	53	28	no	—
88	C₂⁴	C₂×C₄×D₅₃	1	24	yes	—
89	C₂⁴	C₂³×D₅₃	1	7	yes	—
90	C₂²×C₄	—	53	336	no	—
91	C₂²×C₄	—	53	84	no	—
92	C₂²×C₄	—	53	2688	no	—
93	C₂²×C₄	—	53	192	no	—
94	C₂²×C₄	C₂×C₄×D₅₃	1	336	yes	direct product factor
95	C₂²×C₄	C₂³×D₅₃	1	84	yes	—
96	C₄²	—	53	192	no	—
97	C₄²	C₂³×D₅₃	1	64	yes	direct product factor
98	C₂³	—	53	896	no	—
99	C₂³	—	53	336	no	—
100	C₂³	—	53	8	no	—
101	C₂³	C₄²×D₅₃	1	64	yes	direct product factor
102	C₂³	C₂²×C₄×D₅₃	1	84	yes	—
103	C₂³	C₂⁴×D₅₃	1	7	yes	—
104	C₂×C₄	—	53	336	no	—
105	C₂×C₄	—	53	24	no	—
106	C₂×C₄	—	53	768	no	—
107	C₂×C₄	C₂²×C₄×D₅₃	1	336	yes	direct product factor
108	C₂×C₄	C₂⁴×D₅₃	1	24	yes	—
109	C₂²	—	53	448	no	—
110	C₂²	—	53	48	no	—
111	C₂²	C₂×C₄²×D₅₃	1	112	yes	direct product factor
112	C₂²	C₂³×C₄×D₅₃	1	42	yes	—
113	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
114	C₄	—	53	48	no	—
115	C₄	C₂³×C₄×D₅₃	1	48	yes	direct product factor
116	C₂	—	53	32	no	—
117	C₂	C₂²×C₄²×D₅₃	1	28	yes	direct product factor
118	C₂	C₂⁴×C₄×D₅₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂⁴×C₄×D₅₃

C₂

yes

maximal

C₂³×C₄×Dic₅₃

C₂

yes

maximal, complemented

C₂²×C₄²×D₅₃

C₂

yes

maximal, direct product factor

C₂³×C₄²×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂⁵×D₅₃

C₂²

yes

characteristic

C₂⁴×Dic₅₃

C₂²

yes

—

C₂³×C₄×D₅₃

C₄

yes

direct product factor

C₂³×C₄×D₅₃

C₂²

yes

—

C₂²×C₄×Dic₅₃

C₂²

yes

complemented

C₂×C₄²×D₅₃

C₂²

448

yes

direct product factor

C₂⁴×C₄×C₅₃

C₂²

yes

—

C₂²×C₄²×C₅₃

C₂²

yes

complemented

C₂⁴×D₅₃

C₂×C₄

yes

—

C₂⁴×D₅₃

C₂³

yes

—

C₂³×Dic₅₃

C₂×C₄

yes

complemented

C₂³×Dic₅₃

C₂³

yes

—

C₂²×C₄×D₅₃

C₂×C₄

1344

yes

direct product factor

C₂²×C₄×D₅₃

C₂³

336

yes

—

C₂×C₄×Dic₅₃

C₂³

336

yes

complemented

C₄²×D₅₃

C₂³

512

yes

direct product factor

C₂⁵×C₅₃

C₂³

yes

characteristic

C₂³×C₄×C₅₃

C₂×C₄

yes

complemented

C₂³×C₄×C₅₃

C₂³

yes

—

C₂×C₄²×C₅₃

C₂³

112

yes

complemented

C₂⁴×C₅₃

C₂²×C₄

yes

—

C₂⁴×C₅₃

C₂⁴

yes

—

C₂³×D₅₃

C₄²

256

yes

direct product factor

C₂³×D₅₃

C₂²×C₄

336

yes

—

C₂³×D₅₃

C₂⁴

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

336

yes

complemented

C₂²×C₄×C₅₃

C₂⁴

yes

—

C₂²×Dic₅₃

C₂²×C₄

336

yes

complemented

C₂²×Dic₅₃

C₂⁴

yes

—

C₂×C₄×D₅₃

C₂²×C₄

2688

yes

direct product factor

C₂×C₄×D₅₃

C₂⁴

192

yes

—

C₄²×C₅₃

C₂⁴

yes

complemented

C₄×Dic₅₃

C₂⁴

192

yes

complemented

C₂³×C₅₃

C₂×C₄²

yes

complemented

C₂³×C₅₃

C₂³×C₄

yes

—

C₂³×C₅₃

C₂⁵

yes

—

C₂²×D₅₃

C₂×C₄²

896

yes

direct product factor

C₂²×D₅₃

C₂³×C₄

336

yes

—

C₂²×D₅₃

C₂⁵

yes

—

C₂×C₄×C₅₃

C₂³×C₄

336

yes

complemented

C₂×C₄×C₅₃

C₂⁵

yes

—

C₂×Dic₅₃

C₂³×C₄

336

yes

complemented

C₂×Dic₅₃

C₂⁵

yes

—

C₄×D₅₃

C₂³×C₄

768

yes

direct product factor

C₂⁴×C₄²

—

maximal, Sylow

C₂²×C₅₃

C₂²×C₄²

112

yes

complemented

C₂²×C₅₃

C₂⁴×C₄

yes

—

C₂²×C₅₃

C₂⁶

yes

characteristic

C₂×D₅₃

C₂²×C₄²

448

yes

direct product factor

C₂×D₅₃

C₂⁴×C₄

yes

—

C₄×C₅₃

C₂⁴×C₄

yes

complemented

Dic₅₃

C₂⁴×C₄

yes

complemented

C₂⁵×C₄

—

C₂³×C₄²

—

C₂³×C₄²

—

C₂³×C₄²

D₅₃

yes

characteristic, direct product factor

C₂×C₅₃

C₂³×C₄²

yes

complemented

C₂×C₅₃

C₂⁵×C₄

yes

—

D₅₃

C₂³×C₄²

yes

direct product factor

C₂⁶

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

—

C₂⁴×C₄

C₂×D₅₃

yes

—

C₂²×C₄²

—

C₂²×C₄²

—

448

—

C₂²×C₄²

C₂×D₅₃

yes

direct product factor

C₅₃

C₂⁴×C₄²

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂⁵

—

C₂⁵

C₂²×D₅₃

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

1344

—

C₂³×C₄

—

336

—

C₂³×C₄

C₄×D₅₃

yes

direct product factor

C₂³×C₄

C₂²×D₅₃

yes

—

C₂×C₄²

—

336

—

C₂×C₄²

—

512

—

C₂×C₄²

C₂²×D₅₃

112

yes

direct product factor

C₂⁴

—

256

—

C₂⁴

—

336

—

C₂⁴

—

C₂⁴

C₂×C₄×D₅₃

yes

—

C₂⁴

C₂³×D₅₃

yes

—

C₂²×C₄

—

336

—

C₂²×C₄

—

C₂²×C₄

—

2688

—

C₂²×C₄

—

192

—

C₂²×C₄

C₂×C₄×D₅₃

336

yes

direct product factor

C₂²×C₄

C₂³×D₅₃

yes

—

C₄²

—

192

—

C₄²

C₂³×D₅₃

yes

direct product factor

C₂³

—

896

—

C₂³

—

336

—

100

C₂³

—

101

C₂³

C₄²×D₅₃

yes

direct product factor

102

C₂³

C₂²×C₄×D₅₃

yes

—

103

C₂³

C₂⁴×D₅₃

yes

—

104

C₂×C₄

—

336

—

105

C₂×C₄

—

106

C₂×C₄

—

768

—

107

C₂×C₄

C₂²×C₄×D₅₃

336

yes

direct product factor

108

C₂×C₄

C₂⁴×D₅₃

yes

—

109

C₂²

—

448

—

110

C₂²

—

111

C₂²

C₂×C₄²×D₅₃

112

yes

direct product factor

112

C₂²

C₂³×C₄×D₅₃

yes

—

113

C₂²

C₂⁵×D₅₃

yes

characteristic

114

C₄

—

115

C₄

C₂³×C₄×D₅₃

yes

direct product factor

116

C₂

—

117

C₂

C₂²×C₄²×D₅₃

yes

direct product factor

118

C₂

C₂⁴×C₄×D₅₃

yes

—

	subgroup	G/H
center, hypercenter	C₂³×C₄²	D₅₃
commutator, hypocenter	C₅₃	C₂⁴×C₄²
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle	C₂⁵×C₅₃	C₂³
prime radical	C₂⁵×D₅₃	C₂²
cosocle	C₂²×C₅₃	C₂⁶
LCS	G ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄²

subgroup

G/H

center, hypercenter

C₂³×C₄²

D₅₃

commutator, hypocenter

C₅₃

C₂⁴×C₄²

Frattini

C₂²

C₂⁵×D₅₃

Fitting

C₂³×C₄²×C₅₃

C₂

socle

C₂⁵×C₅₃

C₂³

prime radical

C₂⁵×D₅₃

C₂²

cosocle

C₂²×C₅₃

C₂⁶

LCS

G ▷ C₅₃

UCS

C₁ ◁ C₂³×C₄²

Complements

N	H ≅ G/N	type
2. C₂³×C₄×Dic₅₃	C₂	semidirect
3. C₂²×C₄²×D₅₃	C₂	direct (N × H)
4. C₂³×C₄²×C₅₃	C₂	semidirect
7. C₂³×C₄×D₅₃	C₄	direct (N × H)
9. C₂²×C₄×Dic₅₃	C₂²	semidirect
10. C₂×C₄²×D₅₃	C₂²	direct (N × H)
12. C₂²×C₄²×C₅₃	C₂²	semidirect
15. C₂³×Dic₅₃	C₂×C₄	semidirect
17. C₂²×C₄×D₅₃	C₂×C₄	direct (N × H)
19. C₂×C₄×Dic₅₃	C₂³	semidirect
20. C₄²×D₅₃	C₂³	direct (N × H)
22. C₂³×C₄×C₅₃	C₂×C₄	semidirect
24. C₂×C₄²×C₅₃	C₂³	semidirect
27. C₂³×D₅₃	C₄²	direct (N × H)
30. C₂²×C₄×C₅₃	C₂²×C₄	semidirect
32. C₂²×Dic₅₃	C₂²×C₄	semidirect
34. C₂×C₄×D₅₃	C₂²×C₄	direct (N × H)
36. C₄²×C₅₃	C₂⁴	semidirect
37. C₄×Dic₅₃	C₂⁴	semidirect
38. C₂³×C₅₃	C₂×C₄²	semidirect
41. C₂²×D₅₃	C₂×C₄²	direct (N × H)
44. C₂×C₄×C₅₃	C₂³×C₄	semidirect
46. C₂×Dic₅₃	C₂³×C₄	semidirect
48. C₄×D₅₃	C₂³×C₄	direct (N × H)
50. C₂²×C₅₃	C₂²×C₄²	semidirect
53. C₂×D₅₃	C₂²×C₄²	direct (N × H)
55. C₄×C₅₃	C₂⁴×C₄	semidirect
56. Dic₅₃	C₂⁴×C₄	semidirect
60. C₂³×C₄²	D₅₃	direct (N × H)
61. C₂×C₅₃	C₂³×C₄²	semidirect
63. D₅₃	C₂³×C₄²	direct (N × H)
71. C₂²×C₄²	C₂×D₅₃	direct (N × H)
72. C₅₃	C₂⁴×C₄²	semidirect
80. C₂³×C₄	C₄×D₅₃	direct (N × H)
84. C₂×C₄²	C₂²×D₅₃	direct (N × H)
94. C₂²×C₄	C₂×C₄×D₅₃	direct (N × H)
97. C₄²	C₂³×D₅₃	direct (N × H)
101. C₂³	C₄²×D₅₃	direct (N × H)
107. C₂×C₄	C₂²×C₄×D₅₃	direct (N × H)
111. C₂²	C₂×C₄²×D₅₃	direct (N × H)
115. C₄	C₂³×C₄×D₅₃	direct (N × H)
117. C₂	C₂²×C₄²×D₅₃	direct (N × H)

H ≅ G/N

type

2. C₂³×C₄×Dic₅₃

C₂

semidirect

3. C₂²×C₄²×D₅₃

C₂

direct (N × H)

4. C₂³×C₄²×C₅₃

C₂

semidirect

7. C₂³×C₄×D₅₃

C₄

direct (N × H)

9. C₂²×C₄×Dic₅₃

C₂²

semidirect

10. C₂×C₄²×D₅₃

C₂²

direct (N × H)

12. C₂²×C₄²×C₅₃

C₂²

semidirect

15. C₂³×Dic₅₃

C₂×C₄

semidirect

17. C₂²×C₄×D₅₃

C₂×C₄

direct (N × H)

19. C₂×C₄×Dic₅₃

C₂³

semidirect

20. C₄²×D₅₃

C₂³

direct (N × H)

22. C₂³×C₄×C₅₃

C₂×C₄

semidirect

24. C₂×C₄²×C₅₃

C₂³

semidirect

27. C₂³×D₅₃

C₄²

direct (N × H)

30. C₂²×C₄×C₅₃

C₂²×C₄

semidirect

32. C₂²×Dic₅₃

C₂²×C₄

semidirect

34. C₂×C₄×D₅₃

C₂²×C₄

direct (N × H)

36. C₄²×C₅₃

C₂⁴

semidirect

37. C₄×Dic₅₃

C₂⁴

semidirect

38. C₂³×C₅₃

C₂×C₄²

semidirect

41. C₂²×D₅₃

C₂×C₄²

direct (N × H)

44. C₂×C₄×C₅₃

C₂³×C₄

semidirect

46. C₂×Dic₅₃

C₂³×C₄

semidirect

48. C₄×D₅₃

C₂³×C₄

direct (N × H)

50. C₂²×C₅₃

C₂²×C₄²

semidirect

53. C₂×D₅₃

C₂²×C₄²

direct (N × H)

55. C₄×C₅₃

C₂⁴×C₄

semidirect

56. Dic₅₃

C₂⁴×C₄

semidirect

60. C₂³×C₄²

D₅₃

direct (N × H)

61. C₂×C₅₃

C₂³×C₄²

semidirect

63. D₅₃

C₂³×C₄²

direct (N × H)

71. C₂²×C₄²

C₂×D₅₃

direct (N × H)

72. C₅₃

C₂⁴×C₄²

semidirect

80. C₂³×C₄

C₄×D₅₃

direct (N × H)

84. C₂×C₄²

C₂²×D₅₃

direct (N × H)

94. C₂²×C₄

C₂×C₄×D₅₃

direct (N × H)

97. C₄²

C₂³×D₅₃

direct (N × H)

101. C₂³

C₄²×D₅₃

direct (N × H)

107. C₂×C₄

C₂²×C₄×D₅₃

direct (N × H)

111. C₂²

C₂×C₄²×D₅₃

direct (N × H)

115. C₄

C₂³×C₄×D₅₃

direct (N × H)

117. C₂

C₂²×C₄²×D₅₃

direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁴×C₄×D₅₃	2
C₂³×C₄×Dic₅₃	2
C₂²×C₄²×D₅₃	2
C₂³×C₄²×C₅₃	2
C₂⁴×C₄²	53

subgroup

index

C₂⁴×C₄×D₅₃

C₂³×C₄×Dic₅₃

C₂²×C₄²×D₅₃

C₂³×C₄²×C₅₃

C₂⁴×C₄²

Subgroups — C23×C42×D53 #28290640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄²×D₅₃ #28290640